Solve for $X$. $\left[\begin{array}{rr}12 & 8 & -1 \\ -3 & -3 & 7 \end{array}\right]-X=\left[\begin{array}{rr}8 & 6 & -1 \\ -12 & 2 & 10\end{array}\right] $ $X=$
The Strategy First, we can represent the matrices of the equation with letters, which will make the equation easier to handle. Then we can solve the equation for $X$ and obtain an expression with the letters we defined. Finally, we can substitute back the actual matrices into the resulting expression and simplify it. Solving the equation for $X$ We are given the following equation. $\left[\begin{array}{rr}12 & 8 & -1 \\ -3 & -3 & 7 \end{array}\right]-X=\left[\begin{array}{rr}8 & 6 & -1 \\ -12 & 2 & 10\end{array}\right] $ Let's represent the above matrices as follows. $A=\left[\begin{array}{rr}12 & 8 & -1 \\ -3 & -3 & 7 \end{array}\right] ~~~~~~~~~ B = \left[\begin{array}{rr}8 & 6 & -1 \\ -12 & 2 & 10\end{array}\right] $ Then we can rewrite the equation as follows. $A-X=B$ Now it's simple to solve the equation for $X$. $\begin{aligned}A-X&=B\\\\ A&=B+X\\\\ X&=A-B \end{aligned}$ Finding $X$ We found that $X=A-B$. Now we can substitute the actual matrices back into the expression and simplify. $\begin{aligned}X&=A-B \\\\&=\left[\begin{array}{rr}12 & 8 & -1 \\ -3 & -3 & 7 \end{array}\right]-\left[\begin{array}{rr}8 & 6 & -1 \\ -12 & 2 & 10\end{array}\right] \\\\\\&=\left[\begin{array}{rr}(12-8) & (8-6) & (-1+1) \\ (-3+12) & (-3-2) & (7-10)\end{array}\right] \\\\\\&=\left[\begin{array}{rr}4 & 2 & 0 \\ 9 & -5 & -3\end{array}\right]\end{aligned}$ Summary $X=\left[\begin{array}{rr}4 & 2 & 0 \\ 9 & -5 & -3\end{array}\right]$